Stability Theory of Synchronized Motion in Coupled-Oscillator Systems. II: The Mapping Approach

Spatial Structure, Environmental Heterogeneity, and Population Dynamics: Analysis of the Coupled Logistic Map

“…The diagonal need not be locally attracting, however. Yamada and Fujisaka (1983) and Lloyd (1995); proof elaborated in Proposition 1. 4 r is the Lyapunov exponent of the non-spatial logistic map with parameter r Pitchfork bifurcation of strictly out-of-phase two-cycle (II.4) Gyllenberg et al (1993), a Lloyd (1995); b this stabilizes the strictly out-of-phase two-cycle (II.4) and gives rise to the pair of anti-phase two-cycles (II.5)…”

“…Yamada and Fujisaka (1983) and Lloyd (1995) describe the stability criterion for the strictly in-phase orbits, and Gyllenberg et al (1993) and Hastings (1993) give the positions and stability of the various equilibria and period-2 orbits. Lloyd (1995) gave qualitative descriptions of the various complex attractors.…”

“…1). The stability criterion relates the complexity of the nonspatial dynamics, represented by the Lyapunov exponent of f, and the level of dispersal (Yamada and Fujisaka, 1983;Lloyd, 1995; these proofs overlook some subtleties, which we address in Proposition 1). The more complex the non-spatial dynamics, the more dispersal is needed to synchronize the populations.…”

“…They obtain the familiar criterion for the local stability of the strictly in-phase orbits, and analytic expressions for transitions such as the crisis from the outof-phase chaotic attractor to the uncorrelated attractor; the formulas are complicated and not very informative, however. Yamada and Fujisaka (1983) coupled two modified Brusselator models (continuous-time models used to simulate the Belousov Zaboutinsky chemical reaction). They extracted a mapping by taking a Poincare section of each oscillator.…”

Synchronization of nearly identical dynamical systems: Size instability

“…which a single dynamical system is unable to show. Occurrence of synchronization between interacting dynamical units is an important and fundamental nonlinear phenomenon and the study of synchronization of coupled dynamical systems has attracted considerable attention in the past decades [8][9][10][15][16][17][18][19][20][21][22][23]. In particular, the nonlinear behavior of coupled chaotic systems tends to separate the nearby trajectories of the coupled systems, while a suitable coupling between them brings back the trajectories together.…”